Killing disabled and handicapped people was a Nazi policy from early on - Hitler signed the order to begin Aktion T4, as it was named, in 1939, two and a half years before the genocide of Jews got underway. Aktion T4 killed 70,273 adults and about 5000 children in two years of operation. In 1941 the operation was officially closed down, but many of the personnel involved carried on the work unofficially; exact numbers are not available but perhaps as many as 200,000 more people were killed.
The Nazis didn't even attempt to hide their euthanasia programme, although they certainly hid (a) the scale of it (b) the fact that it wasn't voluntary. Hitler actually issued an order in writing - something he normally was careful to avoid when it came to crimes against humanity - directing that "persons who, according to human judgement, are incurable, can, upon a most careful diagnosis of their condition of sickness, be accorded a mercy death."
Aktion T4 operated by asking doctors throughout Germany to provide lists of their long-term patients who were still capable of work. Naively, the doctors attempted to spare their sick patients by being generous in deciding those who were too sick for labour duties - only for this to backfire horribly when SS men in white coats came to take away those patients to 'special treatment centres'. Here they were murdered - at first by lethal injection, later by gassing. The bodies were cremated to hide the evidence, and later on the patient's next of kin would be told that they had tragically died of some infection or disease - pneumonia was the most common excuse used. A fake death certificate was issued to make it all seem above board.
The official justification for Aktion T4 was eugenics, improving the race: to quote Hitler again, "People who are physically and mentally unhealthy or unworthy must not perpetuate the suffering on their children.” In theory, therefore, the programme was aimed at those with hereditary disabilities, which would not include soldiers with crippling war injuries. However, some Nazis openly admitted that another justification was to clear hospital beds for more 'deserving' patients, because during a total war Germany should not be spending money or resources on caring for people who were no use to the State.
It should also be remembered that after Aktion T4 was formally shut down, many Nazi doctors carried right on euthanising the disabled - but now with no central supervision or official orders, acting purely on their own initiative (but safe in the knowledge that nobody in power would care). As such, while I don't know if the story in the book you've read is true, it's certainly conceivable that an over-enthusiastic unit might take it on itself to provide an involuntary 'mercy killing' to seriously wounded soldiers, in order to free resources to treat those who could be cured and sent back to the front.
Sure, it would hurt morale if the soldiers realised what was being done to their wounded comrades; but that was the point. They wouldn't know. The Nazis were good at lying. Good old Hans would be shipped to a field hospital behind the lines - likely with women nurses, the lucky bastard - but then a few weeks later would come a telegram saying that unfortunately, he'd died of pneumonia. How tragic.
